Theorem ringideuNEW 17146

Description: The unit element of a ring is unique.

Hypotheses

Ref	Expression
ringcl.1NEW	|- B = (base` R)
ringcl.3NEW	|- T = (.r` R)

Assertion

Ref	Expression
ringideuNEW	|- (R e. RingNEW -> E!u e. B A.x e. B ((uTx) = x /\ (xTu) = x))

Distinct variable groups:   x,u,B   u,R,x   u,T,x

Proof of Theorem ringideuNEW

Step	Hyp	Ref	Expression
1		eqid 1884	. . . . . 6 |- Struct(3, r, T. ) = Struct(3, r, T. )
2		ringcl.1NEW	. . . . . 6 |- B = (base` R)
3		eqid 1884	. . . . . 6 |- (+g` R) = (+g` R)
4		ringcl.3NEW	. . . . . 6 |- T = (.r` R)
5	1, 2, 3, 4	isringNEW 17142	. . . . 5 |- (R e. RingNEW <-> (R e. Struct(3, r, T. ) /\ (R e. AbelNEW /\ A.x e. B A.y e. B A.z e. B ((xTy) e. B /\ (((xTy)Tz) = (xT(yTz)) /\ (xT(y(+g` R)z)) = ((xTy)(+g` R)(xTz)) /\ ((x(+g` R)y)Tz) = ((xTz)(+g` R)(yTz)))) /\ E.u e. B A.x e. B ((uTx) = x /\ (xTu) = x))))
6	5	simprbi 353	. . . 4 |- (R e. RingNEW -> (R e. AbelNEW /\ A.x e. B A.y e. B A.z e. B ((xTy) e. B /\ (((xTy)Tz) = (xT(yTz)) /\ (xT(y(+g` R)z)) = ((xTy)(+g` R)(xTz)) /\ ((x(+g` R)y)Tz) = ((xTz)(+g` R)(yTz)))) /\ E.u e. B A.x e. B ((uTx) = x /\ (xTu) = x)))
7	6	simp3d 890	. . 3 |- (R e. RingNEW -> E.u e. B A.x e. B ((uTx) = x /\ (xTu) = x))
8		opreq2 4890	. . . . . . . . . 10 |- (x = w -> (uTx) = (uTw))
9		id 73	. . . . . . . . . 10 |- (x = w -> x = w)
10	8, 9	eqeq12d 1899	. . . . . . . . 9 |- (x = w -> ((uTx) = x <-> (uTw) = w))
11	10	rcla4v 2376	. . . . . . . 8 |- (w e. B -> (A.x e. B (uTx) = x -> (uTw) = w))
12		simpl 346	. . . . . . . . 9 |- (((uTx) = x /\ (xTu) = x) -> (uTx) = x)
13	12	ralimi 2168	. . . . . . . 8 |- (A.x e. B ((uTx) = x /\ (xTu) = x) -> A.x e. B (uTx) = x)
14	11, 13	syl5 20	. . . . . . 7 |- (w e. B -> (A.x e. B ((uTx) = x /\ (xTu) = x) -> (uTw) = w))
15		opreq1 4889	. . . . . . . . . 10 |- (x = u -> (xTw) = (uTw))
16		id 73	. . . . . . . . . 10 |- (x = u -> x = u)
17	15, 16	eqeq12d 1899	. . . . . . . . 9 |- (x = u -> ((xTw) = x <-> (uTw) = u))
18	17	rcla4v 2376	. . . . . . . 8 |- (u e. B -> (A.x e. B (xTw) = x -> (uTw) = u))
19		simpr 350	. . . . . . . . 9 |- (((wTx) = x /\ (xTw) = x) -> (xTw) = x)
20	19	ralimi 2168	. . . . . . . 8 |- (A.x e. B ((wTx) = x /\ (xTw) = x) -> A.x e. B (xTw) = x)
21	18, 20	syl5 20	. . . . . . 7 |- (u e. B -> (A.x e. B ((wTx) = x /\ (xTw) = x) -> (uTw) = u))
22	14, 21	im2anan9 622	. . . . . 6 |- ((w e. B /\ u e. B) -> ((A.x e. B ((uTx) = x /\ (xTu) = x) /\ A.x e. B ((wTx) = x /\ (xTw) = x)) -> ((uTw) = w /\ (uTw) = u)))
23		eqtr2 1905	. . . . . . 7 |- (((uTw) = u /\ (uTw) = w) -> u = w)
24	23	ancoms 484	. . . . . 6 |- (((uTw) = w /\ (uTw) = u) -> u = w)
25	22, 24	syl6 25	. . . . 5 |- ((w e. B /\ u e. B) -> ((A.x e. B ((uTx) = x /\ (xTu) = x) /\ A.x e. B ((wTx) = x /\ (xTw) = x)) -> u = w))
26	25	ancoms 484	. . . 4 |- ((u e. B /\ w e. B) -> ((A.x e. B ((uTx) = x /\ (xTu) = x) /\ A.x e. B ((wTx) = x /\ (xTw) = x)) -> u = w))
27	26	rgen2a 2160	. . 3 |- A.u e. B A.w e. B ((A.x e. B ((uTx) = x /\ (xTu) = x) /\ A.x e. B ((wTx) = x /\ (xTw) = x)) -> u = w)
28	7, 27	jctir 317	. 2 |- (R e. RingNEW -> (E.u e. B A.x e. B ((uTx) = x /\ (xTu) = x) /\ A.u e. B A.w e. B ((A.x e. B ((uTx) = x /\ (xTu) = x) /\ A.x e. B ((wTx) = x /\ (xTw) = x)) -> u = w)))
29		opreq1 4889	. . . . . 6 |- (u = w -> (uTx) = (wTx))
30	29	eqeq1d 1892	. . . . 5 |- (u = w -> ((uTx) = x <-> (wTx) = x))
31		opreq2 4890	. . . . . 6 |- (u = w -> (xTu) = (xTw))
32	31	eqeq1d 1892	. . . . 5 |- (u = w -> ((xTu) = x <-> (xTw) = x))
33	30, 32	anbi12d 690	. . . 4 |- (u = w -> (((uTx) = x /\ (xTu) = x) <-> ((wTx) = x /\ (xTw) = x)))
34	33	ralbidv 2123	. . 3 |- (u = w -> (A.x e. B ((uTx) = x /\ (xTu) = x) <-> A.x e. B ((wTx) = x /\ (xTw) = x)))
35	34	reu4 2446	. 2 |- (E!u e. B A.x e. B ((uTx) = x /\ (xTu) = x) <-> (E.u e. B A.x e. B ((uTx) = x /\ (xTu) = x) /\ A.u e. B A.w e. B ((A.x e. B ((uTx) = x /\ (xTu) = x) /\ A.x e. B ((wTx) = x /\ (xTw) = x)) -> u = w)))
36	28, 35	sylibr 217	1 |- (R e. RingNEW -> E!u e. B A.x e. B ((uTx) = x /\ (xTu) = x))

Syntax hints:   -> wi 3   /\ wa 240   /\ w3a 858   T. wtru 1260   = wceq 1298   e. wcel 1300  A.wral 2105  E.wrex 2106  E!wreu 2107  ` cfv 3998  (class class class)co 4884  3c3 7146  Structcstru 16707  basecbs 16758  +_gcplusg 17080  AbelNEWcabel 17084  ._rcmulr 17085  RingNEWcrg 17086

This theorem is referenced by:  ringidcl 17150  ringidmlemNEW 17153

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-tru 1262  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-reu 2111  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-fv 4014  df-opr 4886  df-struct 16708  df-ringNEW 17094

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